Abstract:
By the method of the nonlinear capacity, proposed by S. I. Pokhozhaev in 1997, the blow-up problem for nonlinear partial differential equations of mathematical physics is studied. This method allows to consider the nonlinear systems and high-order equations for the fist time.
In particular, for Korteweg–de Vries equations and its modifications we obtain conditions on the smooth initial functions for the Cauchy problems and initial-boundary conditions for initial-boundary problems, for which smooth solutions blow-up in finite time. Also estimates for blow-up time are presented.
We demonstrate the examples, illustrated the mechanism and properties of the blow-up phenomena. In general, they are similar to the properties of solutions blow-up for the Cauchy problem for the nonlinear (cubic) Schrodinger equation in the three-dimensional space.

References

S. I. Pohozaev, “Blow-up of smooth solutions of the Korteweg–de Vries equation”, Nonlinear Anal., 75:12 (2012), 4688–4698

S. I. Pokhozhaev, “On a class of initial-boundary value problems for equations of Korteweg–de Vries type”, Differ. Equ., 48:3 (2012), 372–378

S. I. Pokhozhaev, “On the nonexistence of global solutions of the Cauchy problem for the Korteweg–de Vries equation”, Funct. Anal. Appl., 46:4 (2012), 279–286